3.2621 \(\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]

[Out]

(2*Sqrt[a]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(Sqrt[1 - e]*Sqrt[a
+ b*x])/Sqrt[a]], -((a*d)/((b*c - a*d)*(1 - e)))])/(b*Sqrt[1 - e]*Sqrt[c + d*x])

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Rubi [A]  time = 0.400461, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(2*Sqrt[a]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(Sqrt[1 - e]*Sqrt[a
+ b*x])/Sqrt[a]], -((a*d)/((b*c - a*d)*(1 - e)))])/(b*Sqrt[1 - e]*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 52.401, size = 83, normalized size = 0.86 \[ \frac{2 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{a d - b c} F\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}\middle | \frac{\left (e - 1\right ) \left (- a d + b c\right )}{a d}\right )}{b \sqrt{d} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

2*sqrt(b*(-c - d*x)/(a*d - b*c))*sqrt(a*d - b*c)*elliptic_f(asin(sqrt(d)*sqrt(a
+ b*x)/sqrt(a*d - b*c)), (e - 1)*(-a*d + b*c)/(a*d))/(b*sqrt(d)*sqrt(c + d*x))

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Mathematica [A]  time = 0.598351, size = 126, normalized size = 1.31 \[ -\frac{2 \sqrt{c+d x} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{(b c-a d) (e-1)}{a d}\right )}{d \sqrt{-\frac{a}{e-1}} \sqrt{\frac{b (e-1) x}{a}+e} \sqrt{\frac{b (c+d x)}{d (a+b x)}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(-2*Sqrt[c + d*x]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticF[ArcSin[Sqrt[-(
a/(-1 + e))]/Sqrt[a + b*x]], ((b*c - a*d)*(-1 + e))/(a*d)])/(d*Sqrt[-(a/(-1 + e)
)]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + (b*(-1 + e)*x)/a])

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Maple [B]  time = 0.171, size = 207, normalized size = 2.2 \[ 2\,{\frac{\sqrt{bx+a}\sqrt{dx+c} \left ( ade-bce+bc \right ) }{ \left ( d{x}^{2}b+adx+bcx+ac \right ) bd \left ( -1+e \right ) }\sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+e \right ) }{a}}}\sqrt{-{\frac{ \left ( dx+c \right ) b \left ( -1+e \right ) }{ade-bce+bc}}}{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x)

[Out]

2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2)*(-(b*x
+a)*(-1+e)/a)^(1/2)*(-(d*x+c)*b*(-1+e)/(a*d*e-b*c*e+b*c))^(1/2)*EllipticF((d*(b*
e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*(a*d*e-b*c*
e+b*c)/((b*e*x+a*e-b*x)/a)^(1/2)/(b*d*x^2+a*d*x+b*c*x+a*c)/b/d/(-1+e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt((a*e + (b*e - b)*x)/a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \sqrt{c + d x} \sqrt{e + \frac{b e x}{a} - \frac{b x}{a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + b*e*x/a - b*x/a)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)), x)